Dressed Hamiltonian

The derivation term is the relative number operator for pairs of photons, one pump-field photon minus one Stokes-field photon. The second term is the well-known RWA Hamiltonian (dressed Hamiltonian used in the usual STIRAP), and the third one can be viewed as a perturbation of this RWA Hamiltonian. 

To describe the shifts and intensities of the m-photon assisted collisional resonances with the microwave field Pillet et al. developed a picture based on dressed molecular states,3 and we follow that development here. As in the previous chapter, we break the Hamiltonian into an unperturbed Hamiltonian H(h and a perturbation V. The difference from our previous treatment of resonant collisions is that now H0 describes the isolated, noninteracting, atoms in both static and microwave fields. Each of the two atoms is described by a dressed atomic state, and we construct the dressed molecular state as a direct product of the two atomic states. The dipole-dipole interaction Vis still given by Eq. (14.12), and using it we can calculate the transition probabilities and cross sections for the radiatively assisted collisions. 

Complex rotation can be usefully applied also to the case of the interaction of an atom with a time-dependent perturbation. With the Floquet formalism by Shirley [41], it was shown that, for a time-periodic field, the dressed states of the combined atom-field system can be characterized non-perturbatively by the eigenstates of a time-independent, infinite-dimensional matrix. The combination of the Floquet approach with complex rotation, proposed by Chu, Reinhardt, and coworkers [37, 42, 43], permits to account for the field-induced coupling to the continuum in an efficient way. As in the time-independent case, this results in complex eigenvalues (this time to the Floquet Hamiltonian matrix) and again the imaginary parts give the transition rate to the continuum. This combination has since then been successfully used to examine various strong field phenomena a review can be found in Ref. [44]. 

It may be noticed that by virtue of being a dressed Hamiltonian including the effect of correlation, diagrams 37-39 account for some of the CIS(D)-type correlation corrections (22-24) to CIS (diagrams 35 and 36). There is no need to make ad hoc adjustments to diagrams to ensure size correctness because diagram 37 (unlinked) cancels exactly between the ground and excited states. In other words, P-EOM-MBPT(2) has the factorization approximation built in. 

At this stage in our discussion it becomes convenient to represent the Hamiltonian and the density operator in a Fourier matrix representation defined by a set of dressed Fourier states n,k In this infinite representation we... 

D. Effective Dressed Hamiltonians by Partitioning of Floquet Hamiltonians... 

Effective Dressed Hamiltonians Partitioning in the Enlarged Space... 

We will establish a precise relation between dressed states in a cavity and the Floquet formalism. We show that the Floquet Hamiltonian K can be obtained exactly from the dressed Hamiltonian in a cavity in the limit of infinite cavity volume and large number of photons K represents the Hamiltonian of the molecule interacting in free space with a field containing a large number of photons. We establish the physical interpretation of the operator... 

For very small field amplitudes, the multiphoton resonances can be treated by time-dependent perturbation theory combined with the rotating wave approximation (RWA) [10]. In a strong field, all types of resonances can be treated by the concept of the rotating wave transformation, combined with an additional stationary perturbation theory (such as the KAM techniques explained above). It will allow us to construct an effective Hamiltonian in a subspace spanned by the resonant dressed states, degenerate at zero field. 

In this subsection we will combine the general ideas of the iterative perturbation algorithms by unitary transformations and the rotating wave transformation, to construct effective models. We first show that the preceding KAM iterative perturbation algorithms allow us to partition at a desired order operators in orthogonal Hilbert subspaces. Its relation with the standard adiabatic elimination is proved for the second order. We next apply this partitioning technique combined with RWT to construct effective dressed Hamiltonians from the Floquet Hamiltonian. This is illustrated in the next two Sections III.E and III.F for two-photon resonant processes in atoms and molecules. 

The second step is the construction of an effective dressed Hamiltonian, independent of the 0-variable, inside the block connected to the initial condition. This can be done by the KAM iterations combined by the RWT techniques to treat the resonances. The second step depends on the specific problem that is treated. 

To extract from the effective Floquet Hamiltonian (185) an effective dressed Hamiltonian independent of 0, we can apply a contact transformation consisting... 

To determine an effective dressed Hamiltonian characterizing a molecule excited by strong laser fields, we have to apply the standard construction of the free effective Hamiltonian (such as the Born-Oppenheimer approximation), taking into account the interaction with the field nonperturbatively (if resonances occur). This leads to four different time scales in general (i) for the motion of the electrons, (ii) for the vibrations of the nuclei, (iii) for the rotation of the nuclei, and (iv) for the frequency of the interacting field. It is well known that it is a good strategy to take into account the time scales from the fastest to the slowest one. 

We first derive the time-dependent dressed Schrodinger equation generated by the Floquet Hamiltonian, relevant for processes induced by chirped laser pulses (see Section IV.A). The adiabatic principles to solve this equation are next described in Section IV.B. 

The preceding analysis is well adapted when one considers slowly varying laser parameters. One can study the dressed Schrodinger equation invoking adiabatic principles by analyzing the Floquet Hamiltonian as a function of the slow parameters. 

---